Numerical solution of some fractional integro-differential equations using Bernstein and Laguerre approximations

Abstract

In this paper, we present a numerical solution of some fractional order integro-differential equations involving the Caputo fractional derivative. The numerical method is based on the use of the Riemann-Liouville fractional integral and the approximation of the unknown solution function by Bernstein and Laguerre polynomials. By using regular collection points, the problem was transformed into a system of linear algebraic equations. The proposed method was tested by solving three problems, where numerical comparisons with other methods indicate the significance of the results in terms of numerical accuracy and ease of application. Furthermore, these results show that the Bernstein approximation is better than the Laguerre approximation. The programming language Mathematica was used to process the numerical results and graphs corresponding to the numerical solution and the resulting errors and show the accuracy of the method.